Differential Geometry and Manifold

by allenlu2007


Differential Geometry 基本上是用微積分於幾何。所以基本要求是 curve or surface 是 continuous and smooth (可微分)。而不處理 singularity 常見於 algebraic geometry.  

幾個基本 concepts (主角和配角).  可以參考本文 

* Manifold:  

就是每個 local point 都可視為是一個近似的 Euclidean space (embedded 在一個 affine space 中).  點和點之間的 transition 是 smooth.  同時相鄰的 two local coordinate charts 也要 smooth.


* Tangent space M and vector space:

from manifold, 每一個 local point p 的近似 Euclidean space (or tangent place) 可以視為 linear vector space.  2D surface in 3D space 的 tangent space TpM: basis (∂s/∂u, ∂s/∂v)


* Metric and first fundamental form:

Local point 附近的 metric.  一般 Euclidean space 的 metric 就是 simple L2 norm.  也就是完全是 flat linear space, curvature = 0.   In general differential geometry 是要處理曲面和曲空間。因此會定義是 bilinear form from metric tensor 

NewImage也稱為 first fundamental form.


如果每點的 bilinear form 都是 positive definite, 稱為 Riemannian geometry.  

? Relativistic geometry (metric = (1, 0; 0, -1)) 是 Riemannian geometry? yes, 在沒有 mass 的情況下,時空是 linear (but (1, 0; 0, -1)) space.  如果有 mass, 則變為 Riemannian geometry?  Wrong, the metric tensor is not positive definite.  稱為 Lorentzian manifold, or pseudo-Riemannian geometry.

* 2nd fundamental form

1st fundamental form 是 focus 在 tangent space or T vector in curve.

2nd fundamental form 則是 focus 在 normal direction or N vector in curve.  明顯就是要定義 curvature.

在 surface 和 curve 最大的不同是 curve 是用 T’ 來定義 normal and curvature.  在 surface 則是用 tangent space 的 cross product 定義 normal and curvature.  首先 normal, then use normal and tangent space 做為座標軸, then Christoffer symbol, 最後 curvature (mean curvature and Gauss curvature) of surface.


以上是 2nd fundamental form.  


* Curvature

在 curve 時的 curvature 在前文已定義。就是用 Frenet framework.  T’ 定義 normal direction 以及 curvature.

在 surface 可以想像所有 normal 方向的切面和 surface 都會有相交曲線。可以定義相對的 curvature.  (Euler) 最後可以找到最大和最小值 curvatures (會是兩個正交的 eigenvector directions), k1 and k2.  可以定義

mean curvature H = k1 + k2 and Gauss curvature K = k1 * k2.




* Covariant derivative and Connection:

Manifold 的不同 points 之間可以定義 connection.   Covariant derivative in direction of a vector field along a tangent vector is defined as the direction derivation with a successive projection onto the tangent plane.

An important issue of the covariant derivation is that it characterises geodesics, i.e. curves on the surface between two points of minimal distance.  

A linear connection 就是 covariant derivative is a mapping:

最常見的是 Affine connection.  為什麼要定義 connection?  簡單來說,vector field (or space) 比較難看出曲面上的 flow.  如果用 connection 更容易看出 (想想電力線和磁力線用 vector field and connection 的差別). 

另外在 Riemannian geometry Levi-Civita connection. 是考慮曲率後的 connection (i.e. not affine connection). 


* Geodesic curves:  在 Riemannian geometry.  Geodesic curve 是任兩點最短距離線。


Some Geometry


1. Euclidean geometry

* Manifold: trivial

* Metric : L2 norm, 沒有曲率 tensor (or unit matrix I)

* Connection: ?

* Geodesic: straight line


2. Relativistic geometry

* Pseudo-Riemannian manifold

* Metric : (1, 0; 0, -1)



3. Information Geometry

* Manifold?

* Metric: Fisher information?

* Connection: alpha 

* D-L divergence